## Convolution Integral Examples

###### Convolution is an important operation since it can be used to compute the output of an LTI system at rest for any input.  The videos below work through a variety of convolution integral examples.

Convolution Integral Example 01 - Convolution of Two Unit Step Functions

10/2/13

Running Time: 6:56

This example computes the convolution of two unit step functions, i.e. y(t) = u(t)*u(t), where * is the convolution operator.

Convolution Integral Example 02 - Convolution of Two Polynomials

4/21/14

Running Time: 9:09

This example computes the convolution of two unit polynomials, x(t) = t^3u(t) and y(t) = t^2u(t), i.e. z(t) = x(t)*y(t), where * is the convolution operator and u(t) is the unit step function.

Convolution Integral Example 03 - Convolution of Two Triangles

10/26/14

Running Time: 13:39

This example computes the convolution of two triangle functions, i.e. y(t) = x(t)*x(t) where x(t) are triangle signals and * is the convolution operator.

The convolution integral is systematically evaluated by sketching the convolution integral integrands for each case of interest as a function of time "t".  Each case provides a portion of the desired convolution for some portion of time.  The final answer for y(t) is a piecewise-defined polynomial in "t".

Convolution Integral Example 04 - Convolution in Matlab (2 Triangles)

11/2/14

Running Time: 7:18

This example also computes the convolution of two triangle functions, i.e. y(t) = x(t)*x(t) where x(t) are triangle signals and * is the convolution operator.  This is the same problem examined in Convolution Integral Example 03.

However, convolution is performed in Matlab in this example.  We show how to appropriately account for the sampling interval (i.e. dt) when using the Matlab convolution operator.

The final convolution plot from Matlab is compared to the analytic results obtained in the previous example, and as expected, they match.

Convolution Integral Example 05 - Convolution Of Unit Step With Pulse

12/15/15

Running Time: 11:02

This example computes the convolution z(t) = x(t)*y(t) where * is the convolution operator, x(t) = u(t) is the unit step function, and y(t) is a finite-width pulse.